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The X axis is infinite. So are the Y and Z axes. Therefore, there must be an infinite number of regular solids. Oh, wait! There's only five. Gee, I guess the mere fact that numbers are infinite doesn't imply that subsets of those numbers are infinite.

That's not actually the argument for why the number of primes is infinite. Rather, assume there are only finitely many prime numbers. Multiply all of them together. Add one to this number. It is easy to show that this number is not divisible by any of the finitely many primes you started with. Hence it must be a prime number as well.

Actually, it doesn't have to be. It only suffices so say, that when you multiply all primes in your list, and add one, you get a number not divisable by any of the number in the list. Hence, one of TWO things can hold true:

1) The number in question really is prime, as you suggested

2) The number in question isn't prime. Then it has prime divisors, none of them in your list (because none of the primes in the list divided our new number).

In both cases, we have derived a way to find at least one new prime fr

IMNSHO, but that was the worst proof of infinite number of primes. Why introduce unique factorizability when you don't need to? Why introduce something foreign that you are not going to prove when there is absolutely no need for it?

The most elegant proof I've seen so far (but I don't know any website showing it, so I can't link to it) is this: For any given N, an integer, consider N!+1, which is greater than N (where N! is defined by N! = 1 * 2 * 3 *... * N). If this number is divisible by no other number than 1 and N!+1, then we are done (i.e. we have proven that given any arbitrary integer, there is a prime greater than tat integer). If this number is divisible by a prime, than that prime can't be less than or equal to N, since any integer (not equal to 1) less than or equal to N divides N! (see the definition of N!) but does not divide 1. Therefore, the prime that divides N! is greater than N. QED.

This proof involves no assumption (additional to assumptions (i.e. axioms) of the set of integers) other than this (which also happens to be much easier to prove than factorizability into primes): if n divides a + b and n divides a, then n divides b as well.

Testing distributed primality algorithms. I should have thought this was obvious.

And, no, one does not encrypt with Mersenne primes. The rarity of such numbers makes a "brute force" crypto-analysis seem rather plausible. Best to use an ordinary prime number-- there are, after all, many more of them to choose from.

No, encryption is out of the question, since it would take enormous computing power to get a new key (see GIMPS). As for the known keys, these are obviously not available for use. A brute force attack would take 21 rounds to complete (on average).

Using a mersenne prime for the public exponent is no problem though. Both 3 and 7 are used quite a lot for this purpose, though 65537 is used most (and this is not a mersenne prime).

Encryption discussions have to take place in a "computing" domain, where a prime only exists if it has been computed to be prime by at least one computer somewhere in the world, and where the prime number can fit on a distribution medium.

Arguing that there are as many Mersenne primes as regular primes is only possible in a theoretical domain in which countably infinite sets can be said to exist.

There ara probably as many Mersenne primes as regular primes. Thus you could just as well encrypt with Mersenne primes.

Not really. Since there are so few known Mersenne primes, the problem of factoring n to find the prime factors p and q to calculate phi(n) in order to crack RSA for example is greatly simplified if either p or q is a Mersenne prime. Perform 42 divisions and you're done.

It's one step closer to proving there's an infinite sequence of these numbers. Just infinity - 42 more to go and the proof is complete!

In all seriousness, they are interesting mainly because they are so simple mathematically that very very early mathematicians got interested in them. But even after hundreds of years of interest among mathematicians, there's no formula for predicting them, and very little successfully proven about them.

Since they are so rare, each find is a significant advancement for those who might be interested in trying to find a pattern.

What use are they?
There may or may not be patterns in the way Merseinne primes occur.
If there are any patterns in Merseinnes, we may need to find more examples than we had before we can find those patterns.
If we do find patterns, they may or may not help us find other patterns that apply to other types of large primes in more general ways.
There is no guarenteed use outside of abstract math, but there is at least a small possibility we could crack one of the really big problems in crypto

Large primes, of around 75-100 digits, are useful in encryption. Huge primes (i.e. over 7 million digits) are not currently useful in themselves, although we certainly learn more about mathemathics and computers as we try to find them.

Finding new Mersenne primes is not likely to be of any immediate practical value. This search is primarily a recreational pursuit. However, the search for Mersenne primes has proved useful in development of new algorithms, testing computer hardware, and interesting young students in math.

It's a mathematical curiosity in some cases - just to find it for the sake of finding it, or for the glory of finding it. You know, like being the first to do something cool.

Also, necessity is the mother of invention. Even if Big Primes aren't really a necessity, it has brought forth some really innovative algorithms and methods to finding prime numbers. Furthermore, it has developed newer and faster ways for multiplying integers.

In 1968, Strassen figured out how to multiply integers quickly by using Fast Fourier Transforms. Strassen, along with Schönhage improved on the method and published a refined version in 1971. GIMPS now uses an improved version of their algorithm. This improved version was developed by Richard Crandall (a longtime researcher of Mersenne Primes).

Another application of finding Prime Numbers is to test computer hardware. Since testing Primes involves a thorough excercise of basic mathematical operations, it provides a good test for hardware. Software routines from GIMPS were used by Intel to test the PII and the Pentium Pro chips before they were shipped. The search for prime numbers was also indirectly responsible for the discovery of the infamous FDIV bug on the Pentium, during the calculation of the twin prime constant (by Thomas Nicely).

Uncommon and unique numbers of varying types are usually useful for mathematics in general. Usually only mathematicians know why.

Whatever the case, this must be a more useful application for CPU power than Seti@home, which is a total waste of energy. Literally.

What we need are more projects that use distributed computing for useful calculations that could further science or solve problems. Universities build giant supercomputers to help their students calculate equations and solve problems. Maybe th

Mersenne primes have two interesting properties that might catch the attention of alien species: when written in binary, they are entirely composed of '1' bits; and, of course, they are prime.

A sure way to prove to another being that you are intelligent is to spew a bunch of numbers which all happen to be prime. The fact that they can be tranmitted using only '1' bits means the modulation is simple -- just send a series of pulses.

Transmitting the same binary signal over and over seems unlikely to impress anyone. You're as likely to be sending a really boring all-white image as a really big prime number.

If anything, anyone receiving the signal will wonder how you managed to build such a powerful transmitter when you haven't discovered binary numbers yet and are apparently using some sort of unary mathematics that really shouldn't work. They're bound to be disappointed when they find out you actually know about "0", but just weren't using it.

No, you have to transmit MULTIPLE numbers.
You transmit the first Mersenne prime. Then you wait for a while. Then you transmit the next one. Wait again. Etc. It's much more efficient to send them in binary than unary (although this most recent prime would require over 2 million bits).

And just because aliens receive a signal with a bunch of strong, equally spaced pulses doesn't mean they'll automatically assume it's intelligent. There are plenty of natural cosmic phenomena which produce equally spaced pul

If it's all 1's (I'll trust you on this, since I haven't RTFA), then it might make sense to transmit the first 42 Mersenne primes by transmitting the number of digits in them instead of transmitting their binary representation. Now, instead of transmitting 2 million 1's, we can transmit 21 0's and 1's. Of course, this gets back to them understanding what we're transmitting. (I think it would take a lot of patience for them to interpret 2 million (give or take) 1's as being a Mersenne prime. And what happens

You can say even more. If M can be written as 2^n - 1, then M is said to be a Mersenne number. If M is also prime, then it is a Mersenne prime. For 2^n - 1 to be a Mersenne prime, n must be a prime number, since we have

If p is a prime number and if 2^p-1 is a Mersenn prime, then, as was pointed out above, 2^(p-1)(2^p-1), is a perfect number. Moreover, if N is an even perfect number, then N can be written (uniquely) as 2^(p-1)(2^p-1) where p is a prime number and 2^p-1 is a Mersenne prime.

Wikipedia [wikipedia.org] has a reasonably intelligible introduction to perfect numbers, and MathWorld [wolfram.com] contains a proof of why every even perfect number must have the form claimed above.

To see why M = 2^(p-1)(2^p-1) is a perfect number when p, 2^p-1 are primes, it suffices to note that s(n), the function that maps an integer to the sum of its divisors (e.g. s(6) = 1 + 2 + 3+6, s(8) = 1 + 2 + 4+8) is multiplicative in the number-theoretic sense, that is to say s(ab) = s(a)s(b) whenever a, b have no prime factors in common. Then evaluating s(M) is simply a case of evaluating it on the factors, which are relatively prime since one is a power of 2, 2^(p-1), and the other is an odd prime, 2^p-1. s(2^p-1) = 2^p-1 + 1 = 2^p (since we have a prime number), and 2^(p-1) = 2^p -1 is an easy formula that is true of all powers of 2. Hence s(M) = 2^p(2^p-1) = 2 ( 2^(p-1) (2^p-1) = 2s(M). That is to say, the sum of all the divisors of M add up to twice M, and if we leave the divisor M itself out of the sum, we see that M is a perfect number.

A Mersenne Prime is where the prime number also fulfills the equation 2^P - 1
2^2 - 1 = 3... 3 is a mersenne prime.
2^3 - 1 = 5... 5 is a mersenne prime.
2^4 - 1 = 7... 7 is a mersenne prime.
The next one is 31 and after that 127. From there they get quite rare (only 42 known).
They are VERY useful in cryptography and quantum physics...both deal with huge numbers. They are also used in some SETI applications because if you wanted to send primes, you'd probably send mersennes as these would be *very*

I'm not sure what else they're actually good for, but searching for these with Prime95 is a great way of putting the flame to your CPU.

Prime95 (which searches for these primes) really puts a load on the CPU and raises the temperature in a hurry. It's commonly used to test the stability of overclocking configurations since it stresses the chip and is able to detect if there is an error in the computation.

Generally, if you can run Prime95 for 24 hours straight, most people will consider the overclocked PC a stable configuration.

I don't know if Prime95 double checks all keys but if it doesn't, that might not be a very nice idea. There'll be a chance of the overclocked CPU doing miscalculation even if it keeps running ok otherwise and you might cause the project to miss a prime.

This has not yet been confirmed, therefore there could be less than 42 known Mersenne primes.

Hovewer, according to MathWorld, there is a chance that it is not the 42nd Mersenne prime at all for another reason:

"However, note that the region between the 39th and 40th known Mersenne primes has not been completely searched, so it is not known if M20,996,011 is actually the 40th Mersenne prime.."
Looks like the big math guys don't exactly know how to count at all;)

Back in the dark ages when I was in university, I took a class called "Mathematics and Poetry". I thought it would be a useful bird course in my senior year, but it turned out to be both interesting and challenging.

As part of the course, we studied Mersenne primes. At the time, I was dabbling in x86 assembler, and I decided to write a program to calculate the then largest known Mersenne prime number: 2^31 - 1, which worked out to 65,050 digits.

The size worked out perfectly, as in 1989 that meant it could fit into one 65KB segment on my blazing-fast 8Mhz 8088. As I recall, the runtime was about two days. The program still works--I can't remember how long it took to run on a 3Ghz P4, but I think it was just a few minutes.

I'm sure any competent programmer (read--not me) could calculate the result much faster, but at the time I was very proud of my little creation.

As part of the course, we studied Mersenne primes. At the time, I was dabbling in x86 assembler, and I decided to write a program to calculate the then largest known Mersenne prime number: 2^31 - 1, which worked out to 65,050 digits.

I don't think it actually did bring back those memories. 2^31-1 is 2147483647. You're thinking of Mersenne prime 31, which is 2^216091 - 1.

I was working through that, too. A number 2 ^ n is certainly not going to have > n digits in decimal. This is given that the minimum value of an n digit integer is 10 ^ (n - 1). One can safely assume that 2 ^ n is smaller than 10 ^ (n - 1) for any integer n > 1.

...waste of time, money and processing power. what kind of use would this have, other than just knowing it? its like winning a eating contest: a completely useless achievement, plus it just turns to poop.

Seventeen or Bust [seventeenorbust.com] (a distributed attack on the Sirpinski Problem), found the fourth largest (well fifth if this one pans out) prime at the beginning of this year, which contains 2,357,207 digits.

Reminds me of when Bart Simpson's 4th-grade class was forced by Principal Skinner to have their annual field trip take place at a box company (instead of the hoped for chocolate factory / fireworks outlet / circus):

Tour guide (speaking in monotone nasal voice): The story of how two brothers (and five other men) parlayed a small
business loan into a thriving paper-goods concern is a long and
interesting one. And, here it is: it all began with the filing of form
637/A, the application for a small business or farm...

Counterexample! The Mersennes (2^n-1) and the primes form intersecting sets with their intersection being, of course, the Mersenne primes. The primes are not, as you stated, a subset of the Mersennes, nor is the opposite case true.

Factoring a large prime number, we'll call it n, is extremely easy. Its factors are n and 1, by definition. Factoring large, 'hard' composite numbers (ones that don't have obvious factors like 2 or 3, there are some other criteria as well) is the difficult problem.

Nah, the goal is obviously to find a technologically inferior bunch of intelligent aliens, and bring our cookbooks to their planet. Before any vegans complain, I want to point out that anything from another planet can't technically be an animal, so morally they should be fine to eat. And spicy!